Question

How do you find all the asymptotes for function `f(x) = (11(x-5)^5(x+6)^5)/ (x^10+11x^5+30)`?

Answer 1

`x=-root(5)6` and `x=-root(5)5` are two vertical asymptotes. Horizontal asymptote is `y=11`.

Explanation:

To find asymptotes of `f(x)=(11(x−5)^5*(x+6)^5)/(x^10+11x^5+30)`, we should factorize denominator `(x^10+11x^5+30)`

The factors of `(x^10+11x^5+30)` are `(x^5+5)` and `(x^5+6)`

As `(x+a)` is a factor of `(x^5+a^5)` (as `-a` is zero of latter)

two factors of `(x^10+11x^5+30)` are `x=-root(5)6` and `x=-root(5)5`

and hence `x=-root(5)6` and `x=-root(5)5` are two vertical asymptotes.

It is also apparent that the highest degree of numerator is `11x^10` and that of denominator is `x^10`. As their ratio is `11`

Horizontal asymptote is `y=11`.